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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras whose only ideals are and itself. Throughout the article, unless otherwise stated, is a finite-dimensional Lie algebra over a field of characteristic 0. The following conditions are equivalent: * is semisimple *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate, * has no non-zero abelian ideals, * has no non-zero solvable ideals, * The radical (maximal solvable ideal) of is zero. == Examples == Examples of semisimple Lie algebras, with notation coming from classification by Dynkin diagrams, are: * , the special linear Lie algebra. * , the odd-dimensional special orthogonal Lie algebra. * , the symplectic Lie algebra. * , the even-dimensional special orthogonal Lie algebra. These Lie algebras are numbered so that ''n'' is the rank. Except certain exceptions in low dimensions, many of these are simple Lie algebras, which are ''a fortiori'' semisimple. These four families, together with five exceptions (E6, E7, E8, F4, and G2), are in fact the ''only'' simple Lie algebras over the complex numbers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「semisimple lie algebra」の詳細全文を読む スポンサード リンク
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